ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfrnf GIF version

Theorem dfrnf 4842
Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfrnf.1 𝑥𝐴
dfrnf.2 𝑦𝐴
Assertion
Ref Expression
dfrnf ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrn2 4789 . 2 ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2 nfcv 2306 . . . . 5 𝑥𝑣
3 dfrnf.1 . . . . 5 𝑥𝐴
4 nfcv 2306 . . . . 5 𝑥𝑤
52, 3, 4nfbr 4025 . . . 4 𝑥 𝑣𝐴𝑤
6 nfv 1515 . . . 4 𝑣 𝑥𝐴𝑤
7 breq1 3982 . . . 4 (𝑣 = 𝑥 → (𝑣𝐴𝑤𝑥𝐴𝑤))
85, 6, 7cbvex 1743 . . 3 (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
98abbii 2280 . 2 {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10 nfcv 2306 . . . . 5 𝑦𝑥
11 dfrnf.2 . . . . 5 𝑦𝐴
12 nfcv 2306 . . . . 5 𝑦𝑤
1310, 11, 12nfbr 4025 . . . 4 𝑦 𝑥𝐴𝑤
1413nfex 1624 . . 3 𝑦𝑥 𝑥𝐴𝑤
15 nfv 1515 . . 3 𝑤𝑥 𝑥𝐴𝑦
16 breq2 3983 . . . 4 (𝑤 = 𝑦 → (𝑥𝐴𝑤𝑥𝐴𝑦))
1716exbidv 1812 . . 3 (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
1814, 15, 17cbvab 2288 . 2 {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
191, 9, 183eqtri 2189 1 ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1342  wex 1479  {cab 2150  wnfc 2293   class class class wbr 3979  ran crn 4602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-br 3980  df-opab 4041  df-cnv 4609  df-dm 4611  df-rn 4612
This theorem is referenced by:  rnopab  4848
  Copyright terms: Public domain W3C validator